ML DP progr

ML DP progr - 1 Developing an MOR/ML Dynamic Programming...

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1 Developing an MOR/ML Dynamic Programming Program At this point, we have modeled a variety of dynamic programming models and solved them both by hand and by use of the MOR/DP support package. It is now time to consider the details of the computational mechanics of dynamic programming. Therefore, we will revisit some of these example problems and develop programs to implement their solution in MOR/ML. There are two levels of accomplishment possible. First, being able to develop a program to solve a specific fixed length problem requires a certain amount of knowledge about the process. Then to actually generalize the program to an n -dimensional problem based on the problem data requires a bit more programming and MOR/ML manipulation skills. We begin by restating the limited-inventory production-planning example problem of the introduction. Problem. The demand for a product over the next four time periods is 2, 3, 4, and 2 units. The cost of placing an order is $15, independent of the number of units ordered. The individual item cost is $100, and the holding is $2 per unit per period. Due to the company accounting system, holding costs are charged on the inventory units at the beginning of a period; that is, on entering inventory. A maximum of four units can be held from period to period. The company orders from a local supplier, so we can consider that units ordered in a period are available for use in the same period. However, due to the limited production capacity of the supplier, orders are limited to once a period with a maximum of five units. We want the optimal ordering policy that meets the demands at minimal cost for the four time periods. The company currently has no items in inventory and has no requirements for a specified number in inventory at the end of the planning horizon. Since demands must be met, ordering for future demand is the only viable option for decreasing costs. Solution. We have a four period decision problem with production quantities x i to be determined. The inventory at the beginning of each period is represented by the variable s i , with the initial inventory, s 1 , set to zero. The period demand for units is defined as d i . The relationships between these variables are: s i + 1 = s i + x i d i 0,1,2,3,4 { } ,
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2 where { } 0, 1, 2, , 5 i x " and { } , 4 i s " . Then the cost function Cost ( s i ,x i ,d i ) is defined for feasible s i +1 ( s i + 1 = s i + x i d i ) as Cost ( s i , x i , d i ) = 2 s i + 15( x i > 0) + 100 x i , s i + x i d i 0, 1, 2, 3, 4 { } , +∞ , otherwise. That is, if the output inventory level of period i , s i +1 , is not feasible then we assign an infinite cost as the function value for that case. The sequence of optimization problems to solve this multiple period planning problem uses f i (s i ) as the minimal cost of meeting the demands from period i to the end of the problem (until period 4) given the entering inventory level into period i is s i units.
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This note was uploaded on 10/16/2010 for the course ISEN 689 taught by Professor Klutke during the Spring '07 term at Texas A&M.

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ML DP progr - 1 Developing an MOR/ML Dynamic Programming...

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